IX] CRITICAL BEHAVIOUR OP STRUCTURES 215
It has been found that the critical loading for a uniform
prismatic strut of negligible weight under end thrusts and pin-
PI
jointed at the ends is given by F = rP -~-, where I = length of
strut, / the least moment of inertia, and E = Young's modulus,
so that cj> for this structure is IT. For a strut under the same
conditions but with fixed ends <f> — 2?r.
In certain cases of simple shapes of structures, it is obvious
that the function <fi2 may be derived by calculation as for example
in the case of the prismatic beam supported either on simple or
clamped supports, but for more complicated problems where the
calculation is too abstruse a number of points on the characteristic
curve may be derived by a series of tests to destruction on
models.
§ 12. In one other type of structure at least other than that
of a simple strut the crippling coefficient <f> can be evaluated. The
case referred to, of extreme importance in aeronautical stress
calculations, is that which corresponds to the spars of a wing or
the longerons of a fuselage. For purposes of calculation these
members maybe supposed, with sufficient accuracy for the present,
to consist of a strut held by simple supports at a number of points
along its length loaded laterally in any given manner and under
the influence of end thrust varying from bay to bay in a manner
determined by the general configuration and loading of the rest
of the structure and inclinations and positions of attachment of
wires to each bay. The actual values of these end thrusts must of
course be calculated according to the principles enunciated in the
previous chapter.
§ 13. Case of a longeron ofn Bays. Let there be n + 1 supports,
figured 0, 1,2, ...,n, and spaced as in the diagram, so that the Hh
bay appears to the left of the support numbered r. Let the
bending moments at these supports be MQ, M1? ..., Mn, the
deflections as already defined in the previous chapter Sl} 82> •••>
8n~i, and suppose the lateral loading considered uniform along
each bay, Zl3 Z2, ...,Zn, is wl9w29 ...9wn. E^, E2I2, ..., Enln
are as before the flexural rigidities of the various bays and FI9
F2,..., Fn the thrusts in these members.
Applying the generalised form of the equation of three moments.